Question: The equation of hyperbola $H$ is $\dfrac {(x+9)^{2}}{81}-\dfrac {(y-4)^{2}}{16} = 1$. What are the asymptotes?
Explanation: We want to rewrite the equation in terms of $y$ , so start off by moving the $y$ terms to one side: $\dfrac {(y-4)^{2}}{16} = - 1 + \dfrac {(x+9)^{2}}{81}$ Multiply both sides of the equation by $16$ $(y-4)^{2} = { - 16 + \dfrac{ (x+9)^{2} \cdot 16 }{81}}$ Take the square root of both sides. $\sqrt{(y-4)^{2}} = \pm \sqrt { - 16 + \dfrac{ (x+9)^{2} \cdot 16 }{81}}$ $ y - 4 = \pm \sqrt { - 16 + \dfrac{ (x+9)^{2} \cdot 16 }{81}}$ As $x$ approaches positive or negative infinity, the constant term in the square root matters less and less, so we can just ignore it. $y - 4 \approx \pm \sqrt {\dfrac{ (x+9)^{2} \cdot 16 }{81}}$ $y - 4 \approx \pm \left(\dfrac{4 \cdot (x + 9)}{9}\right)$ Add $4$ to both sides and rewrite as an equality in terms of $y$ to get the equation of the asymptotes: $y = \pm \dfrac{4}{9}(x + 9)+ 4$